Mind Reading with Bases

by Spencer Bowen

In Mind Reading with Math we showed how you could use binary to read someone’s mind.  

For that activity, we used a set of cards representing the first four binary place values: 8, 4, 2, 1.

Copy of Color Counting (11-19 Visit)

Binary only has two digits: “1” or “0” (which could also be thought of as “yes” or “no”).  This made our mind reading trick pretty easy.  If a number we were trying to “mind read” appeared on the card corresponding to a particular place value, then that place value was required as an addend of the number we’re trying to “mind read.”  So if you’re reading a volunteer’s mind, and they respond “Yes, yes, no, and yes,” then you add together the place values of the cards showing their number and get 13.

This trick can extend to higher numbers by including more cards, with each card doubling the possible numbers someone could chose from.  So if you added a card, you would now have a “16s” place value and could choose from 32 possible numbers (0-31).

Mind Reading with Ternary 

We can extend this trick to base 3 or ternary, but we will need to rethink how we make our cards.  Our binary cards just needed two digits: “on the card” = 1 and “not on the card” = 0.  Ternary has three possible digits (0, 1, 2), so we cannot use the simple off/on distinction.

Instead, we can use symbols to represent our digits.  For example, let “Circle” = 0, “Square” = 1, and “Diamond” = 2.  Using colors to represent place values like before, we get:

Now Mind Reading involves a bit more mental math.  For each card, we ask our volunteer what shape their number is on the card, then multiply the corresponding digit by the card’s place value.  

For example, let’s say a volunteer chooses “50.” Then they would respond “Square, Diamond, Square, Diamond.”

This method works for all numbers 0 – 80.  The only information the mind reader has to memorize is each digit’s shape and each place value’s color.

Even More Bases 

This activity generalizes nicely to any base.  You just have to decide how you want to distinguish your place values and digits.  Here are base 4 or quaternary cards using our same color scheme for place values and the four playing card suits as our digits.

In fact, it’s surprisingly easy to make mind reading cards for different bases!  If you look at the ternary and quaternary cards, you can notice some pretty distinct geometry.  The “ones” place value has the digits in a repeating pattern.  Looking at the above example, you can see on the “1s” card (which is red) that “hearts” = 0, “spades” = 1, “diamonds” = 2, and “clubs” = 3.  And the suits keep repeating in a cycle.

Next, the “fours” place value (which is orange) has a cycling pattern as well (“hearts” , “spades”, “diamonds”, “clubs”); the only difference is that each digit is repeated four times in a row before moving to the next digit in the cycle. This pattern continues with the “sixteens” place value as well, except with each digit being repeated sixteen times before moving to the next digit in the cycle.

This geometry generalizes to any base assuming you lay out the numbers in reading order.  For each card corresponding to a particular place value, the symbols will form into sets of increasing numbers sharing the same symbol.  The length of these sets will be to the card’s place value.  The symbol of these groups will then cycle between your digits’ symbols in order.

How to make your own base cards

  1. Choose your favorite base.  Call it b.
  2. Decide the largest number you want your volunteers to be able to choose.  This number has to be equal to one less than a power of b.
  3. Make a number of cards equal to the power you chose in step 2.  Make a way to distinguish between your cards.  The examples shown above color-code the cards. But you can use different methods as long as you can tell the cards apart.
  4. Lay your numbers out on the cards in increasing order from left to right, making new rows as you run out of room (start with 0 in the top left corner and end with your largest number in the bottom right corner).
  5. Determine which symbols represent your digits.  They should be distinct and easy to see.
  6. For your “ones” place value card, label each of your bases’ digits with the correct symbol.  These symbols then repeat in a cycle all the way to your largest number.
  7. Your next card should represent the b place value.  Label the first b (0, …, b-1) numbers with your “0” symbol, then label the next set of b numbers with your “1” symbol.  Keep cycling through your digits’ symbols, labeling sets of numbers with a symbol before moving onto the next symbol.
  8. For each card representing some place value b^k, repeat step 7 but instead label sets of numbers of length b^k with the same symbol before moving onto the next digit.

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